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Page 1: Unstable manifolds of recurrent flows in pipe flowonline.itp.ucsb.edu/online/transturb-c17/budanur/pdf/Budanur_Trans… · By acting with H(x), we are able to bring stability eigenvectors

Unstable manifolds of recurrent flowsin pipe flow

Nazmi Burak Budanur

January 10, 2017

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Motivation

E2E1

E0

x

y

z

Lorenz system:

x = σ(y − x) ,

y = ρx − y − xz ,

z = xy − bz ,

Unstable manifolds shape theattractor.

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MotivationPlane Couette flow (Gibson, Halcrow, Cvitanovic (2008))

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The problem

• Continuous symmetries → relative invariant solutions

• Can be avoided in invariant subspaces of plane Couette flow

• Cannot be avoided in pressure-driven flows (pipes, channels)

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Relative equilibria and relative periodic orbits

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Symmetry reduction

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Pipe flow

• Navier-Stokes equations for fluctuations around base flow:

uτ + uHP · ∇u+ u · ∇uHP + u · ∇u = −∇p + 32β

Rez+

1

Re∇2u

• β = β(τ) ensures constant-flux

• Incompressibility: ∇ · u = 0

• No-slip on the wall: u(z , θ, r = D/2) = 0

• Periodic in axial and azimuthal directions:u(z , θ, r) = u(z + L, θ, r) and u(z , θ, r) = u(z , θ + 2π, r)

Navier-Stokes equations along with incompressibility and boundaryconditions induces a finite-time flow

x(τ) = f τ (x(0)),

where x is the state space vector corresponding to a point in thespace of allowable velocity fields.

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Symmetries

• Translation:

• Rotation:

• Reflection:

gz(l)u(z , r , θ) = u(z − l , r , θ)

gθ(φ)u(z , r , θ) = u(z , r , θ − φ)

σ[u, v ,w ](z , r , θ) = [u, v ,−w ](z , r ,−θ)

• Equivariance under SO(2)z ×O(2)θ:

G = {gz(l), gθ(φ), σ}

• In state space:

gf τ (x) = f τ (gx) , where g ∈ G

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Symmetry reduction

• Each solution x has infinitely many “symmetry copies” on its“group orbit”:

Mgx = {g x | ∀g ∈ G}

• Symmetry reduction is a coordinate transformation x → xsuch that each group orbit Mgx is represented by a singlepoint x .

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SO(2) symmetry reduction: First Fourier mode slice

Project the solution x(τ) onto a subspace where group orbits arecircular, then fix the polar angle on this projection:

Let x ′ be a state space vectorcorresponding to the velocity field

[u′, v ′, w ′](z , θ, r) = J0(αr) cos(2πz/L)

then

a1 = 〈x , x ′〉 , a2 = 〈x , gz(−L/4)x ′〉

x = gz(−l)x , l =L

2πarg a1 + ia2

x(τ2)

x(τ1) φ(τ1)

a1x(τ1) x(τ2)

a2

φ(τ2)

x(τ0)

Budanur et al. (2015) PRL, Willis, Short, Cvitanovic (2016) PRE

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Why call it a slice?

“Polar” coordinatetransformation defines acodimension-1 “slice hyperplane”in state space:

〈x(τ)−x ′, t ′z〉 = 0 , 〈tz(x), t ′z〉 > 0

M

x(τ)

x(τ)x ′

t ′z

x(0)

• tz(x) = Tz x : Group tangent

• Tz : Generator of infinitesimal translations i.e. gz(l) = e lTz

• t ′z = tz(x ′) : Slice tangent

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Projecting perturbations

Small perturbations δx to x canbe brought to the slice by thetransformation

δx =

(1− tz(x)⊗ t ′z〈tz(x), t ′z〉

)gz(−l)δx

U(x) = 0

x

x + δx

xx + δx

gz(−l)

gz(−l − δl)

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Projection operator

H(x) = 1− tz(x)⊗ t ′z〈tz(x), t ′z〉

allows us to project tangent space onto the slice.

Example: If x = v(x) = limδτ→0 fδτ (x)/δτ then

˙x = v(x) = v(x)− 〈t′z , v(x)〉〈tz(x), t ′z〉

tz(x)

is the symmetry reduced state space velocity.

By acting with H(x), we are able to bring stability eigenvectors oftraveling waves and Floquet vectors of relative periodic orbits tothe slice.

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Unstable manifolds of traveling waves

• “Asymmetric” traveling wave, S1

• Invariant under “shift-and-reflect”

xS1 = σgz(L/2)xS1

• Lies on laminar-turbulent boundary

• Has two real unstable (µ > 0)stability eigenvalues

Here, computed at Re = 3000 for a pipeof length L = 5D.

(Pringle & Kerswell 2007, Mellibovsky & Meseguer 2007)

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Unstable manifold of S1

xφ(τ = 0) = xS1+ε

(V1

µ1cosφ+

V2

µ2sinφ

), e1,2 = 〈xφ(τ), V1,2〉

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Unstable manifold of S1

xφ(τ = 0) = xS1+ε

(V1

µ1cosφ+

V2

µ2sinφ

), e1,2 = 〈xφ(τ), V1,2〉

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Unstable manifold of S1

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Unstable manifold of S1

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Unstable manifold of S1

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Recurrences on the unstable manifold

τ = 0 τ = 250 τ = 330

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A new traveling wave on the “edge”

Initial condition:

Converged solution:

λ1,2 = 0.131± 0.178iλ3,4 = 0.095± 0.066i

λ5 = 0.073

x(0) = xNew ± εV5

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“Almost” complete picture

λ1,2 = 0.131± 0.178i

λ3,4 = 0.095± 0.066i

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“Destabilizing” turbulence

Kuhnen et al. (2017) to appear

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Too much turbulence kills turbulenceSO(2)z × SO(2)θ-reducedmanifold:

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Too much turbulence kills turbulence

N:

τ = 0 τ = 35 τ = 70

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Localized relative periodic orbits

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Poincare section

xp

v(xp)

〈xP − xp, v(xp)〉 = 0

〈v(xP), v(xp)〉 > 0

U(xP) = 0

x

x + δx

xPxP + δxP

f τ (x)

f τ+δτ (x + δx)

δxP =

(1− v(xP)⊗ v(xp)

〈v(xP), v(xp)〉

)df τ (x)

dxδx

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Unstable manifold of the lower branch at Re= 1700

-0.03 -0.01 0.01 0.03e1

-0.015

-0.005

0.005

0.015

e 2 LBUB

xP(δ) = xp,P ± εΛδ1V1,P , δ ∈ (0, 1]

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Heteroclinic connection

0 5 10 15 20n

0.0

0.02

0.04

0.06

0.08

0.1

||aP[n

]−aUB,P||

UB:

n = 0 (LB):

n = 9 (closest approach):

n = 20:

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On the same scale

n = 0 (LB):

n = 9 (closest approach to UB):

n = 20:

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Puff formation from a random initial perturbation

-0.03 -0.01 0.01 0.03e1

-0.015

-0.005

0.005

0.015

e 2 LBUB

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Flow structures

0 5 10 15 20n

0.0

0.02

0.04

0.06

0.08

0.1||a

P[n]−aUB,P||

UB:

n = 0:

n = 4 (closest approach):

n = 20:

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Conclusions

• We computed and visualized unstable manifolds of travelingwaves and relative periodic orbits

• We found a new traveling wave on the laminar-turbulentboundary of a small pipe

• We found strong numerical evidence of a heteroclinicconnection between localized relative periodic solutions ofpipe flow

Acknowledgements:

• Ashley Willis (openpipeflow.org)

• Predrag Cvitanovic

• Yohann Duguet

• Genta Kawahara

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Complex conjugate eigenvalues

λ1,2 = µ± iω

x(0) = xTW + εe2πµωδRe V1 , δ = (0, 1]

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Projection

U(x) = 0

x

x + δx

xx + δx

f φ(x)

f φ+δφ(x + δx)

δx =

(1−

∂θfθ(x)|θ=φ ⊗∇U(x)

〈∂θf θ(x)|θ=φ, ∇U(x)〉

)∇f φ(x)δx